Amenability, Critical Exponents of Subgroups and Growth of Closed Geodesics

TL;DR

This paper establishes a connection between the amenability of quotient groups and the equality of critical exponents and growth rates of closed geodesics in convex co-compact groups acting on Hadamard manifolds, extending classical results.

Contribution

It proves that a normal subgroup has the same critical exponent as the parent group if and only if the quotient is amenable, and similarly relates growth rates of geodesics to amenability.

Findings

Critical exponent equality characterizes amenability of quotient groups.

Growth rate of closed geodesics matches the critical exponent under certain conditions.

Extends classical results of Kesten and Brooks to geometric group actions.

Abstract

Let $\Gamma$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold $X$. We show that a normal subgroup $\Gamma_0$ has critical exponent equal to the critical exponent of $\Gamma$ if and only if $\Gamma / \Gamma_0$ is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $X / \Gamma$. These statements are analogues of classical results of Kesten for random walks on groups and of Brooks for the spectrum of the Laplacian on covers of Riemannian manifolds.